Digital Signal Processing Reference
In-Depth Information
From time-domain description to power-spectral description.
Here is an example
of how the preceding concepts are applied in communication systems. In digital
communications, the received sequence can often be represented by the equation
∞
y
(
n
)=
c
(
m
)
s
(
n
−
m
)+
q
(
n
)
,
(E
.
37)
m
=
−∞
where
s
(
n
) is a transmitted vector sequence,
c
(
n
) is the impulse response of a
MIMO channel, and
q
(
n
) is additive noise. It is typical to assume that
s
(
n
)and
q
(
n
) are jointly wide sense stationary processes. Given any other process
u
(
n
)
,
we can write
∞
y
(
n
)
u
†
(
n
m
)
u
†
(
n
k
)+
q
(
n
)
u
†
(
n
−
k
)=
c
(
m
)
s
(
n
−
−
−
k
)
.
m
=
−∞
If
u
(
n
) is jointly WSS with all other processes under discussion, then we can take
expectations on both sides to get
∞
R
yu
(
k
)=
c
(
m
)
R
su
(
k − m
)+
R
qu
(
k
)
.
(E
.
38)
m
=
−∞
Taking Fourier transforms, we get
S
yu
(
e
jω
)=
C
(
e
jω
)
S
su
(
e
jω
)+
S
qu
(
e
jω
)
.
(E
.
39)
Using
z
-transform notation, this is sometimes written as
S
yu
(
z
)=
C
(
z
)
S
su
(
z
)+
S
qu
(
z
)
.
(E
.
40)
Such equations serve as starting points for finding optimal prefilters and equalizers
that minimize the reconstruction error. Many such examples can be found in this
topic.
E.3 Cyclo WSS processes
Imagine we convert a wide sense stationary (WSS) discrete-time signal
s
(
n
)into
a continuous-time signal
x
(
t
) as in Fig. E.1. Then
x
(
t
) is a “cyclo WSS process”
with period
T,
rather than a WSS process. In this section we shall explain what
this term means. For such processes the power spectrum can only be defined in a
time-average sense, as elaborated below. With the power spectrum appropriately
defined, we will see that the
average power spectrum
of
x
(
t
)isgivenby
S
xx
(
jω
)=
1
|
2
S
ss
(
e
jωT
)
,
T
|
F
(
jω
)
(E
.
41)
where
S
ss
(
e
jω
) is the power spectrum of
s
(
n
)
.
Thus the
power
at the channel
input takes the form
∞
−∞
|
1
T
|
2
S
ss
(
e
jωT
)
dω
2
π
p
0
=
F
(
jω
)
(
E.
42)
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